arXiv:cond-mat/0110486

Lower energy bounds for quantum lattice Hamiltonians

Dean Lee, Nathan Salwen

Published Oct 23, 2001• 0 citations

Abstract

We derive general lower energy bounds for the ground state energy of any translationally invariant quantum lattice Hamiltonian. The bounds are given by the ground state energy of renormalized Hamiltonians on finite clusters.

DOI: 10.48550/arxiv.cond-mat/0110486

Related papers

A variational lower bound on the ground state of a many-body system and the squaring parametrization of density matrices

F. Uskov, O. Lychkovskiy

A variational upper bound on the ground state energy E gs of a quantum system, E gs ⩽ 〈Ψ| H |Ψ〉, is well-known (here H is the Hamiltonian of the system and Ψ is an arbitrary wave function). Much less known are variational lower bounds on the ground state. We consider one such bound which is valid for a many-body translation-invariant lattice system. Such a lattice can be divided into clusters which are identical up to translations. The Hamiltonian of such a system can be written as , where a term H i is supported on the i 'th cluster. The bound reads , where is some wisely chosen set of reduced density matrices of a single cluster. The implementation of this latter variational principle can be hampered by the difficulty of parameterizing the set , which is a necessary prerequisite for a variational procedure. The root cause of this difficulty is the nonlinear positivity constraint ρ > 0 which is to be satisfied by a density matrix. The squaring parametrization of the density matrix, ρ = τ 2 /tr τ 2 , where τ is an arbitrary (not necessarily positive) Hermitian operator, accounts for positivity automatically. We discuss how the squaring parametrization can be utilized to find variational lower bounds on ground states of translation-invariant many-body systems. As an example, we consider a one-dimensional Heisenberg antiferromagnet.

arXiv:1902.09246Journal of Physics Conference SeriesFeb 1, 20199 citations

Lower Bounds on Ground-State Energies of Local Hamiltonians through the Renormalization Group

Ilya Kull, Norbert Schuch, Benjamin Dive +1 more

Given a renormalization scheme, we show how to formulate a tractable convex relaxation of the set of feasible local density matrices of a many-body quantum system. The relaxation is obtained by introducing a hierarchy of constraints between the reduced states of ever-growing sets of lattice sites. The coarse-graining maps of the underlying renormalization procedure serve to eliminate a vast number of those constraints, such that the remaining ones can be enforced with reasonable computational means. This process can be used to obtain rigorous lower bounds on the ground-state energy of arbitrary local Hamiltonians by performing a linear optimization over the resulting convex relaxation of reduced quantum states. The quality of the bounds crucially depends on the particular renormalization scheme, which must be tailored to the target Hamiltonian. We apply our method to 1D translation-invariant spin models, obtaining energy bounds comparable to those attained by optimizing over locally translation-invariant states of n≳100 spins. Beyond this demonstration, the general method can be applied to a wide range of other problems, such as spin systems in higher spatial dimensions, electronic structure problems, and various other many-body optimization problems, such as entanglement and nonlocality detection. Published by the American Physical Society 2024

arXiv:2212.03014Physical Review XApr 9, 202424 citations

Derivation of a Lower Bound on the Free Energy

Robert H. T. Yeh

For a Hamiltonian with a ground-state energy E0 = 0, we show that for any normalized state |ψ〉, 〈ψ| e−βH |ψ〉≤e−β〈ψ|H|ψ〉a, where a=exp [−β 〈ψ| H 2 |ψ〉/〈ψ| H |ψ〉]. From this inequality, a general formula for a lower bound on the free energy is derived.

Journal of Mathematical PhysicsMay 1, 19704 citations

Lower Bounds for the Ground-State Energies of the 2D Hubbard and t-J Models

Joachim Stolze, Roser Valentí, P. J. Hirschfeld

We present simple lower bounds on the ground-state energy of the two-dimensional (2D) Hubbard and t-J models for arbitrary values of band filling and coupling constant. For the Hubbard model we derive two types of bounds, both based on decomposing the model Hamiltonian into a sum of sub-Hamiltonians. For a decomposition into local cluster sub-Hamiltonians, we perform a generalized Legendre transform on previously derived bounds for the grand-canonical potential. For a decomposition into spin-up and spin-down parts, previous results on the spinless Falicov-Kimball model may be used to obtain bounds for the Hubbard model, generalizing a result of Langer and Mattis to arbitrary filling. For the 2D t-J model we have only considered the decomposition into clusters. The 1D Hubbard model is used as a test case. The bounds may be improved by diagonalizing the Hamiltonian for larger clusters.

Jan 1, 19915 citations

Lower bounds to variational problems with guarantees

Jens Eisert

Variational methods play an important role in the study of quantum many-body problems, both in the flavor of classical variational principles based on tensor networks as well as of quantum variational principles in near-term quantum computing. This work stresses that for translationally invariant lattice Hamiltonians with periodic boundary conditions, one can easily derive efficiently computable lower bounds to ground state energies that can and should be compared with variational principles providing upper bounds. As small technical results, it is shown that (i) the Anderson bound and a (ii) common hierarchy of semidefinite relaxations both provide approximations with performance guarantees that scale as a constant in the energy density for cubic lattices. (iii) Also, the Anderson bound is systematically improved as a hierarchy of semidefinite relaxations inspired by the quantum marginal problem.

arXiv:2301.06142Physical Review AJan 6, 20265 citations

The usefulness of restrictions for variational ground state lower bounds of a simple quantum spin lattice

Joel H. Sebold, Jerome K. Percus

We utilize reduced density matrices and restrictions upon them to obtain variational lower bounds to the ground state energy of the transverse Ising model. A systematic use of restrictions on a cluster representation (like 3 or 4 sites) of the system in 1, 2, and infinite dimensions, shows that a small class of conditions is sufficient to yield good results for little effort. The technique is analytically straightforward, and is clearly a promising direction to go for ground state energy bounds of quantum lattices with short range interactions.

Molecular PhysicsNov 1, 19952 citations